Spectral sequences for Hochschild cohomology and graded centers of derived categories

TL;DR

This paper interprets the characteristic homomorphism from Hochschild cohomology to the graded center as an edge homomorphism in a spectral sequence, explaining its injectivity and surjectivity failures through various examples.

Contribution

It provides a spectral sequence framework to understand the characteristic homomorphism in Hochschild cohomology and graded centers, with illustrative examples.

Findings

Spectral sequence interpretation clarifies the homomorphism's properties.

Examples include modules over dual numbers and coherent sheaves.

Explains failures of injectivity and surjectivity in specific cases.

Abstract

The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded center of its homology category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence. This gives a conceptual explanation of the failure of the characteristic homomorphism to be injective or surjective, in general. To illustrate this, we discuss modules over the dual numbers, coherent sheaves over algebraic curves, as well as examples related to free loop spaces and string topology.